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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 388080.y
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.y1 | 388080y7 | \([0, 0, 0, -179702127003, 29320837462877002]\) | \(1826870018430810435423307849/7641104625000000000\) | \(2684305014134413824000000000000\) | \([2]\) | \(1528823808\) | \(5.0477\) | |
| 388080.y2 | 388080y6 | \([0, 0, 0, -11405801883, 443173968921418]\) | \(467116778179943012100169/28800309694464000000\) | \(10117492105596405572173824000000\) | \([2, 2]\) | \(764411904\) | \(4.7011\) | |
| 388080.y3 | 388080y4 | \([0, 0, 0, -3088929963, 5802244171738]\) | \(9278380528613437145689/5328033205714065000\) | \(1871727577552037438225879040000\) | \([2]\) | \(509607936\) | \(4.4983\) | |
| 388080.y4 | 388080y3 | \([0, 0, 0, -2157361563, -30029878803638]\) | \(3160944030998056790089/720291785342976000\) | \(253037086380161868251529216000\) | \([2]\) | \(382205952\) | \(4.3545\) | |
| 388080.y5 | 388080y2 | \([0, 0, 0, -2024038443, -34908345659558]\) | \(2610383204210122997209/12104550027662400\) | \(4252304598315199469774438400\) | \([2, 2]\) | \(254803968\) | \(4.1518\) | |
| 388080.y6 | 388080y1 | \([0, 0, 0, -2021780523, -34990416987302]\) | \(2601656892010848045529/56330588160\) | \(19788824740379555266560\) | \([2]\) | \(127401984\) | \(3.8052\) | \(\Gamma_0(N)\)-optimal |
| 388080.y7 | 388080y5 | \([0, 0, 0, -995273643, -70366370515238]\) | \(-310366976336070130009/5909282337130963560\) | \(-2075919253297303607433423912960\) | \([2]\) | \(509607936\) | \(4.4983\) | |
| 388080.y8 | 388080y8 | \([0, 0, 0, 8915478117, 1850556729369418]\) | \(223090928422700449019831/4340371122724101696000\) | \(-1524763831895904893646607220736000\) | \([2]\) | \(1528823808\) | \(5.0477\) |
Rank
sage: E.rank()
The elliptic curves in class 388080.y have rank \(0\).
Complex multiplication
The elliptic curves in class 388080.y do not have complex multiplication.Modular form 388080.2.a.y
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
